# Calculating unweighted performance of stocks within a period

The well known calculation of unweighted index of stocks is just calculating an arithmetic average.

And then, to calculate the performance of the index, I calculate the %change of the unweighted index between 2 points of time.

But how about I calculate the %change for every stocks between 2 points of time first, and then take the arithmetic average of those %changes?

Is the second method valid? I found the results are quite different. Which one is better?

## 1 Answer

The first method gives you the return of a a price-weighted average, like the Dow Jones average. So I suppose it is OK to use.

The second method gives you a rebalanced EW (equal weighted) average: you initially invest the same amount (say 1000 dollars) in each stock and then you rebalance to equal weights at each point where you do the calculation. Obviously the results depend on how frequently you do the calculation; if it is very frequently (e.g. daily) it becomes somewhat unrealistic because it means you would have to buy/sell every stock every day to get back to equal dollars per stock, if one stock now has too many dollars you must sell some of it and buy more of another that has gone down in relative price. It can be done but it is an active strategy, not a passive index. With monthly or quarterly (or yearly) recalculation it is probably OK, in the sense that there is a more realistic amount of activity involved. Still I would recommend that you either NOT USE this method, or at least disclose the calculation period (e.g. EW rebalanced monthly) otherwise no one will be able to reproduce your calculation.

But the most popular indexes like the S&P index are of a third kind: market weighted indexes.

Or else, the fourth method: to compute the return to an investor you must know the number of shares of each stock that the investor owns. This is the most general method. The others are a special case of this: for example the first method assumes that the investor buys an equal number of shares in each company, which is a strange method of investing, to say the least. The second method simulates constantly buying and selling shares, as I described above.

• But why it is so different between 1st and 2nd method? I compare both under same period (e.g. also 1m perforance) and assume they all invested by 1 share. – roy104 Aug 30 '15 at 12:03
• The second is an active trading strategy, not the continuous holding of the same number of shares. By averaging the returns in each month you are assuming that you have the same dollar amount invested in each stock at the beginning of each month. But how is that possible given that the stocks grew at different rates the PREVIOUS month?! Only by assuming that on the last day of the previous month you sold/bought some stocks to REBALANCE your holdings. – Alex C Aug 30 '15 at 12:56
• Just consider two stocks and two holding months, no trading allowed. The first stock stays constant in price: 10, 10 , 10. The second stock increases: 10, 20, 30. In the first month it is correct to average the returns. In the second month it is wrong: you have twice as much in stock 2 as stock 1, so you can't take a simple average of returns. – Alex C Aug 30 '15 at 13:07